# Dot Product with Zero Vector is Zero

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## Theorem

Let $\mathbf u$ be a vector quantity.

Let $\cdot$ denote the dot product operator.

Then:

- $\mathbf u \cdot \mathbf 0 = 0$

where $\mathbf 0$ denotes the zero vector.

## Proof

By definition of dot product:

\(\ds \mathbf u \cdot \mathbf 0\) | \(=\) | \(\ds \norm {\mathbf u} \norm {\mathbf 0} \cos \theta\) | Definition of Dot Product | |||||||||||

\(\ds \) | \(=\) | \(\ds \norm {\mathbf u} \times 0 \times \cos \theta\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 0\) |

$\blacksquare$

## Sources

- 1951: B. Hague:
*An Introduction to Vector Analysis*(5th ed.) ... (previous) ... (next): Chapter $\text {II}$: The Products of Vectors: $2$. The Scalar Product